function [J, grad] = linearRegCostFunction(X, y, theta, lambda)
%LINEARREGCOSTFUNCTION Compute cost and gradient for regularized linear 
%regression with multiple variables
%   [J, grad] = LINEARREGCOSTFUNCTION(X, y, theta, lambda) computes the 
%   cost of using theta as the parameter for linear regression to fit the 
%   data points in X and y. Returns the cost in J and the gradient in grad

% Initialize some useful values
m = length(y); % number of training examples

% You need to return the following variables correctly 
J = 0;
grad = zeros(size(theta));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost and gradient of regularized linear 
%               regression for a particular choice of theta.
%
%               You should set J to the cost and grad to the gradient.
%
hx = X * theta;
diff = hx - y;
J = diff' * diff + lambda * theta(2:end)' * theta(2:end);
J = 0.5 * J / m;

grad(1) = sum(diff .* X(:, 1));
grad(1) = grad(1) / m;
n = size(X, 2);
for i = 2 : n
    grad(i) = sum(diff .* X(:, i)) + lambda * theta(i);
    grad(i) = grad(i) / m;
end









% =========================================================================

grad = grad(:);

end
